Regularity Estimates for Quasilinear Elliptic Equations with Measure Data and Matrix-Valued Weight

Abstract

This paper is concerned with the existence and local gradient regularity of solutions to a quasilinear elliptic Dirichlet problem with finite Radon measure and a degenerate matrix-valued weight. Under a small log-BMO condition on $\mathbb M$, we first prove the existence of a SOLA (solution obtained by a limit of approximations) and then establish a local Calder\'on-Zygmund estimate for $|\mathbb{M}|^{\frac{1}{p-1}}\mathbb{M} Du$ in terms of the fractional maximal function $\mathcal{M}_1(\mu)^{\frac{1}{p-1}}$.